Conditional multinomial resampling central limit theorem

Let ${\mathcal{F}}_N$ contain the corrected particles and their weights. By [F2], conditionally on ${\mathcal{F}}_N$ the selected values $Y_1^N,\dots ,Y_N^N$ are IID with law ${\omega }_t^N$. Their centered conditional variance is $s_N^2={\omega }_t^N({\varphi }^2)-{\omega }_t^N(\varphi {)}^2$, which converges in probability to $s^2={\mathrm{Var}}_{{\pi }_t}(\varphi )$ by [F1] and [F3].

Put $m_N={\omega }_t^N(\varphi )$ and ${\xi }_i^N=\varphi (Y_i^N)-m_N$. By [F3], $m_N\to {\pi }_t\varphi$ in probability, hence $m_N=O_P(1)$. For a fixed threshold $a>0$, choose fixed $K$ and then $R\ge K$. On $\{\mathrm{\mid }{m}_N\mathrm{\mid }\le K\}$ and for all sufficiently large $N$ with $a\sqrt{N}>R+K$, the event $\mathrm{\mid }\varphi (Y_i^N)-m_N\mathrm{\mid }>a\sqrt{N}$ implies $\mathrm{\mid }\varphi (Y_i^N)\mathrm{\mid }>R$, and $\mathrm{\mid }\varphi (Y_i^N)-m_N{\mathrm{\mid }}^2\le 4\varphi (Y_i^N{)}^2$ there. Thus the conditional Lindeberg sum is at most $4{\omega }_t^N({\varphi }^2{\mathbf{1}}_{\{\mathrm{\mid }\varphi \mathrm{\mid }>R\}})$ on that event. For fixed $R$, [F3] makes this upper bound converge to $4{\pi }_t({\varphi }^2{\mathbf{1}}_{\{\mathrm{\mid }\varphi \mathrm{\mid }>R\}})$, which tends to zero as $R\to \mathrm{\infty }$ by the finite target second moment. Given any error probability, first choose $K$ so that the probability of $\mathrm{\mid }{m}_N\mathrm{\mid }>K$ is small, then choose $R\ge K$ with this tail integral small, and finally let $N\to \mathrm{\infty }$. This proves that the conditional Lindeberg sum tends to zero in probability using only target $L^2$.

Along any subsequence, choose a further subsequence on which $s_N\to s$ almost surely. If $s>0$, then eventually $s_N>s\mathrm{/}2$; the normalized Lindeberg sum for draws divided by $s_N\sqrt{N}$ is bounded by step 2.1 at threshold $as\mathrm{/}2$, divided by $s_N^2$. It therefore tends to zero in probability, and a further subsequence makes that convergence almost sure. Apply [F4] to obtain a standard-normal limit for the normalized row. Slutsky [F6] with $s_N\to s$ gives convergence of the conditional characteristic function of $N^{-1\mathrm{/}2}{\sum }_i{\xi }_i^N$ to $e^{-u^2{s}^2\mathrm{/}2}$. If $s=0$, conditional [F5] gives $P(\mathrm{\mid }{N}^{-1\mathrm{/}2}{\sum }_i{\xi }_i^N\mathrm{\mid }>a\mid {\mathcal{F}}_N)\le \min (1,s_N^2\mathrm{/}{a}^2)\to 0$, giving the degenerate characteristic-function limit. The subsequence criterion therefore gives convergence in probability of these conditional characteristic functions along the full sequence.

The conditional characteristic functions are bounded by one, so their differences from $e^{-u^2{s}^2\mathrm{/}2}$ converge to zero in $L^1$. For arbitrary real $v,u$, multiply the conditional resampling characteristic function at $u$ by the bounded factor $e^{iv\sqrt{N}({\omega }_t^N(\varphi )-{\pi }_t(\varphi ))}$ and take expectations. The resulting joint characteristic function tends to $e^{-v^{2}V\mathrm{/}2}{e}^{-u^2{s}^2\mathrm{/}2}$. Thus the resampling fluctuation is asymptotically independent of the weighted fluctuation, and their sum has variance $V+s^2$.